We study the Plateau problem with a lower-dimensional obstacle in ${ℝ}^n$. Intuitively, in ${ℝ}^3$ this corresponds to a soap film (spanning a given contour) that is pushed from below by a “vertical” 2D half-space (or some smooth deformation of it). We establish almost optimal $C^{1,\frac{1}{2}-}$ estimates for the solutions near points on the free boundary of the contact set, in any dimension $n\ge 2$. The $C^{1,\frac{1}{2}-}$ estimates follow from an ε-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi’s improvement of flatness. To prove it, we follow Savin’s small perturbations method. A nontrivial difficulty in using Savin’s approach for minimal surfaces with thin obstacles is that near a typical contact point the solution consists of two smooth surfaces that intersect transversally, and hence it is not very flat at small scales. Via a new “dichotomy approach” based on barrier arguments we are able to overcome this difficulty and prove the desired result.

中文翻译：

---

具有低维障碍的最小曲面的规则性

我们研究了具有较低维障碍的高原问题 ${ℝ}^{ñ}$。直观地${ℝ}^3$这对应于一块肥皂膜（跨越给定的轮廓），该肥皂膜由“垂直”二维半空间（或其一些平滑变形）从下方推动。我们建立几乎最佳的$C^{\mathrm{1个}，\frac{\mathrm{1个}}{2}--}$ 在任意维度上的接触集自由边界上的点附近的解的估计 $ñ\ge 2$。的$C^{\mathrm{1个}，\frac{\mathrm{1个}}{2}--}$根据De Giorgi改善平整度的精神，采用ε-正则结果估算出最小的带有薄壁障碍物的表面。为了证明这一点，我们遵循Savin的小扰动方法。在没有障碍物的最小表面上使用Savin方法的一个不小的困难是，在典型的接触点附近，该解决方案由两个相交的光滑表面组成，因此，在小范围内不是很平坦。通过基于障碍论证的新“二分法”，我们能够克服这一困难并证明所需的结果。